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LIBRARY OF CONGRESS. 



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UNITED STATES OF AMEEICA. 




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JUNIOR COURSK 



IN 



Mechanical Drawling, 



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BY 



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Director of the Drawing School of the Franklin Institute of Philadelphia. 



PUBLISHED BY 

WILLIAMS & BROWN, 
N. E. Cor. Chestnut and Tenth Sts., Philadelphia. 



Entered according to Act of Congress, in the year 1888, by 

WILLIAM H. THORNE, 
In the office of the Librarian of Congress, at Washington. 



/ 3S3 



The want of harmony between the modern practice of draughting in our best 
Engineering Establishments and the theories and methods given in all text-books 
on the subject, has led me to endeavor to present the subject of Mechanical 
Drawing in such a manner that the student^s train of thought, method of manipu- 
lation, and knowledge of conventionalities will correspond with what experience has 
approved and adopted. For this purpose I have arranged three progressive courses 
of instruction : — A Junior Course, treating of Lines, Plane Surfaces, and single 
Solids containing only plane surfaces : — An Intermediate Course, treating 
of Solids with Curved Surfaces, the Intersection of Solids and the Development of 
their Surfaces : — and a Senior Course, treating of the application of these prin- 
ciples to the making of Working Drawings, and of the technicalities, style, 
and conventionalities of practical draughting. 

In the following pages, comprising the Junior Course, those plane prob- 
lems only which are actually useful, are considered, and the endeavor has been 
to suggest methods best adapted for ease, rapidity, accuracy, and clearness. 

WM. H. THORNE. 

Go WEN Ave., Mount Airy, 
Philadelphia, 1888. 



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OUTFIT AND PREPARATION. 

In pursuing this Course of Mechanical Drawing, the student should be 
provided with the following outfit : 

Drawing Board, at least 21 J in. by 18| in. Drawing Paper in 
sheets 21 in. by 16 in. Four Thumb Tacks for fastening paper to 
board. T-square with blade at least 21 in. long. One 45° Triangle, 
6 in. One 30° and 60° Triangle, 7J in. Irregular Curve. Extra 
hard Pencil. File for sharpening leads. Scales of full, half, and quarter 
sizes, graduated to ^ in. the entire length. One 3J-in. and one 5J-in. 
Compasses, with pen, pencil, and needle-points. Siberian Leads, HHHHHH, 
for use in Compasses. One Small Spring Bow or Spacing Dividers. One 
medium Ruling Pen, Writing Pen, Rubber. One bottle each of Liquid India 
Ink, Red Ink and Blue Ink, if for use in a class, although fine India Ink, 
Carmine, and Prussian Blue, to be mixed as required, are preferable but in- 
convenient for school purposes. 

The Pencil should be sharpened with a pen-knife to a point IJ in. 
long, for f in. of which the lead should be exposed. The lead is then 
sharpened to a flat point about 32" i^* wide for strength and durability, by 
rubbing it on the file. The lead in the Compasses are sharpened in the same 
manner. The file should be convenient for constant use in keeping the pencils 
in good working condition. 

5 



6 JUNIOR COURSE. 

To prepare the drawing paper for the exercises, find by measurement the 
centre of the sheet vertically in the following manner: Place the Scale flat 
on the paper with its zero at the lower edge of the latter, and make a 
short, fine mark with the pencil opposite the 12-in. graduation of the scale, 
then shift the zero to this mark, see Avhat graduation comes opposite the 
upper edge of the paper, add this dimension to the 12 in., divide the sum 
by two, and, after bringing the zero back to the lower edge of the paper, 
make a mark opposite the dimension thus found, which will be the centre 
of the sheet vertically. 

In laying off dimensions, always do it with the Scale and Pencil. Never 
set the Divider points to the graduations on the Scale and prick the paper with 
theniy as the habit is ruinous to the scale besides being an uncertain and clumsy 
method. 

Through the central point thus found, draw a horizontal line by holding 
the head of the T-square against the left-hand edge of the Board and mov- 
ing the pencil from left to right in contact with the upper edge of the blade. 

Find the centre of the sheet horizontally in a similar manner, and 
through this centre draw a vertical line, by holding the T-square as before 
and, against its top edge, sliding the short square side of the 30° 
Triangle, until the long square side coincides with the desired point and 
using the latter side to guide the pencil, always keeping this side to the left 
and moving the pencil aw^ay from the person. The T-square and Triangle 



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MECHANICAL DRAWING. 7 

are shifted up and down to make any length of vertical line required, but 
the edge of the T-square itself should never be used to guide the pencil for 
vertical lines if it is possible to avoid it. 

These precautions limit the requirements for accuracy of work to a Board 
with its left-hand edge straight, sl T-square with the inner edge of its head 
and the upper edge of its blade straight, a Triangle with two of its edges 
straight and square with each other and the other edge making angles of ex- 
actly 30° and 60° with the other sides, and a Triangle with two of its 
edges straight and square and the other . edge at exactly 45°. 

From the central horizontal line, lay off 7J in. above and below, and 
from the central vertical line, lay off 10 in. to the right and left, which 
will give the margin lines 15 by 20 in. and four spaces, each 7 J by 10 
in. These spaces, or subdivisions of them are the standard to be used in 
all the studies in this Course. 

PLATE I. 

Fig. 1 is an exercise in horizontal and vertical parallel lines, the hori- 
zontal being drawn by using the T-square and the vertical by using the 
Triangle sliding on the T-square, all the lines to be J in. apart. The 
different kinds are designated : a, fine lines ; b, shade lines ; c, dotted lines ; 
d, the long and short dot; e, the long dot. 



8 JUNIOR COURSE. 

Fig. 2 shows inclined parallel lines, produced by sliding one of the 
square edges of the Triangle along the top edge of the T-square, the inclined 
edge of the Triangle being used to guide the pencil, the T-square being in 
the position required to bring the edge of the Triangle to the desired incli- 
nation. Lines to be drawn J in. apart and the kind to be varied at dis- 
cretion. 

To ink these and all other straight lines, use the Ruling Pen, filling it 
with ink by means of a writing pen, thus avoiding the necessity of wiping 
the outside. When, however, the Ruling Pen is filled by dipping it in the 
ink, the outside must be wiped off before using. It must also always be 
wiped out between the blades before laying it down and frequently during 
use. For this purpose, a rag should be kept convenient at all times as a 
part of the operation of inking, the same as a file is kept as a part of that 
of penciling. Keeping the instruments in good working condition is import- 
ant, but trying to save ink or pencil is trifling. 

The margin and division lines of the sheet are to be Black. All lines 
in Figs. 1 and 2 are to be black, except the dimension lines and the triangles 
in Fig. 2, which are to be Red. All Red lines are to be fine. 

In general, Hed ink is used for all lines which are merely explanatory or 
imaginary. 

Dimensions are to be in Black ink, feet being expressed by ft., inches by 
'', and degrees by °. 



MECHANICAL DRAWING. 9 

The arrow-heads or pointers at the extremities of dimension lines must 
be small and black, and must always touch the line to which they extend, 
because they are intended to indicate definite size and not merely direction. 

All the figures on the plates are drawn one-fi^urth size, or to a 
scale of 3 in. to the foot, but should be drawn full size by the student. 
Each Plate represents a sheet of drawing-paper 21 in. by 16 in. 

Fig. 3. To ered a perpendicular to a given horizontal line at a given 
point on the line. 

Length of line, 4 in. ; distance of point from right-hand end, 1 J in. 

Draw a horizontal line with the T-square and lay off a length of 4 
in. upon it. At a distance of IJ in. from the right-hand mark, 
make another mark for the point, through which draw a vertical line, 
holding the Triangle against the T-square so that the pencil, guided by the 
Triangle, will pass through the point. This will be the required line. 

Fig. 4. To drop a perpendicidar to a given horizontal line from a 
given point outside the line. 

Draw a 4-iu. horizontal line with the T-square. Assume any point 
above it and through this point draw a vertical line as before. 

If the given line were vertical it would be drawn with the Triangle an(J 
the required line with the T-square. 

Fig. 5. To erect perpendiculars at the extremities of a horizontal line.. 



10 JUNIOR COURSE. 

Draw a horizontal line, lay off 4 in. upon it and erect a perpendi- 
cular at the left-hand point, as in Fig. 3. In erecting the perpendicular at 
the rfglit-hand point, Fig. 5 shows the Triangle turned over in the opposite 
direction, that is, witii its vertical edge to the right, contrary to the instruc- 
tions already given. This is done to call attention to the importance of always 
using it in the proper manner, unless the location of the required line renders 
it inconvenient or impossible to do so. When it must be used in this manner, 
the pencil or pen, which is guided by it should be moved toward the person. 

Fig. 6. To erect a perpendicular to a given inclined line at a given jooint. 

This is an i.njxjrtant operation to be thoroughly understood and practiced, 
because of its great simplicity and usefulness. 

The method is based on the fact that if one of the square edges of a 
right-angled Triangle is held against any ruler, and a line is drawn along 
the hypothenuse and then the Triangle is revolved and its other square edge 
is held against the same ruler, a line drawn along the hypothenuse in this 
last position will be perpendicular to the one drawn in the first position, 
provided that the ruler has not meanwhile been moved. 

Draw a line of any inclination and upon it lay ofP the dimensions given 
in the Fig. Bring the inclined edge of either Triangle to coincide with the 
line and hold it down firmly. Bring the edge of the other Triangle or of 
the T-square, whichever can be used most conveniently, in contact with one 
of the square edges of the first Triangle and hold it down firmly. Eevolve 



MECHANICAL DEAWING. 11 

the first Triangle to bring its other square edge in contact and slide it until 
its inclined edge coincides with the given point, when a line drawn along the 
inclined edge through the point will be the perpendicular required. 

Note the distinction between a vertical line and a line which is perpen- 
dicular to another line. 

Fig. 7. To bisect a given line or to divide it into two equal parts. 

Draw a 4-in. line, and upon it lay off any distance greater than one- 
half its length, say 2J in., from one end. Place the needle-point of the 
Compasses at this (nd and set the pencil-point to the mark thus laid off and 
describe an arc of a circle. Remove the needle-point to the other end of the 
line, and describe a similar arc, intersecting the first. A perpendicular to the 
line drawn through these intersections will bisect the line. 

This method is used only for lines which are quite short. Long lines 
are bisected by means of the scale in the manner described on page 8. 

Fig. 8. To bisect a given angle. 

Draw any two lines intersecting each other to represent the given angle. 
Place the needle-point of the Compasses in this intersection, and with any arc 
intersect both lines. From these intersections describe arcs of radius greater 
than half the distance between them. Then a line drawn through the inter- 
sections of these arcs will bisect the angle, and should be perpendicular to 
one through the first intersections. 



12 JUNIOR COURSE. 

Fig. 9. To bisect an are of a circle. 

On a line lay oif 2f in. as the diameter of a circle by making a 
mark at the zero of the scale, one at If in, and one at 2f in. Put 
the needle-point of the Compasses at the If -in. mark on the line, and set 
the pencil-point so that it will cut both the other marks, or will average 
any inaccuracy in the measurement. Describe the circle, and on it mark two 
points to limit an arc. From these points describe arcs intersecting each 
other, and a line drawn through their intersections will bisect the arc and 
also its chord. It will be perpendicular to the chord, and will pass through 
the centre of the circle, and consequently will be a radius. 

Fig. 10. To find the centre of a circle which will pass through three 
given points^ a, 6, and c. 

Connect a and 6, and b and c, by lines. Bisect these lines and the inter- 
section of the bisecting lines will be the centre of a circle which will pass 
through the given points. 

PLATE 2. 

Fig. 11, To draw a tangent to a circle at a given point, a. 

Draw a circle 2 in. diameter and assume any point on the circum- 
ference, as a. Draw a radius through this point. A perpendicular to this 
radius, drawn through the point a, will be the tangent required. 



MECHANICAL DRAWING. 13 

Fig. 12. To draio a tangent to a circle from any given outside 'point , a, 
and to find the exact point of tangency. 

Draw a circle 2 in. diameter and assume any outside point, as a. 
From this point draw a line just touching the circle. Draw a perpendicular 
to this line to pass through the centre of the circle. The point of inter- 
section of the perpendicular with the tangent line and circle will be the 
point of tangency. 

Fig. 13. To draw a tangent to a given circle at a given point j a, when 
the centre of the circle is lost or cannot be used. 

Draw an arc of a circle of 3 J in. radius, assume any point on it, 
as a. With this point as centre and any radius, say IJ in., intersect the 
arc in two points equidistant from a, and connect these two points by a line. 
A parallel to this line (as in Fig. 2), drawn through a will be the tangent 
required. 

Fig. 14. To draw a circle tangent to another circle at a given pointy a. 

Draw a 2J-in. circle. Through the given point a, on the circumfer- 
ence, draw a radius and on this, produced, describe a circle IJ in. diameter, 
touching the first circle at the given point. 

Fig. 15. To draw a circle tangent to two other circles. 
Draw a circle 2 in. diameter and another If in. diameter, with its 
centre 2 J in. from the centre of the first, and let it be required to 



14 JUNIOR COURSE. 

draw a IJ-in. circle which will be tangent to both of these. From the 
centre of the first circle describe an arc of radius equal to the sum of the 
radii of the first circle and the tangent circle, or If in. Intersect this 
arc with another described from the centre of the second circle, of radius equal 
to the sum of the radii of the second circle and tangent circle, or IJ in. 
The intersection of these arcs will be the centre of the tangent circle. 

Fig. 16. Angles are measured by equal divisions, called degrees, of a circle 
whose centre is at the intersection of the lines forming the angle. 

The circle is divided into 360°, and the zero can be taken at 
either extremity of either the horizontal or vertical diameters. By means of 
the two triangles already provided, an angle of 15°, or any multiple thereof, 
can be drawn. 

Draw a 3-in. circle and its horizontal and vertical diameters and mark 
the extremities of the former 0°, and of the latter 90°. Draw two diameters 
with the 45° Triangle in its two diiferent positions. Draw four diameters 
with the 30° Triangle in its four different positions. Draw four other diameters 
with the 30° and 45° Triangles, both combined in four different positions. 
These diameters will divide the circle into arcs of 15° each. 

Fig. 17. In Trigonomet?^, angles are dealt with by means of certain lines, 
called the functions of the circle, a graphical illustration of which is given in 
this figure. 



MECHANICAL DRAWING. 15 

Draw a 3-in. circle, a horizontal diameter ha, and a vertical radius 
eg. From the centre of the circle draw an indefinite line chj making any 
angle, say 30°, with ha, then hca will be the angle, and tlie difference 
between this and 90° or hcg, will be the complement of the angle. 
From the point d, where ch intersects the circle, draw a horizontal line, dfy 
and a vertical line, de. From a draw a vertical line ah, and from g a 
horizontal line gh. Then ca=Radius, ab^ Tangent, de=Sine^ df=^Co-siney 
cb=Secant, ch=z Co-secant, and gh= Co-tangent. 

Fig. 18. In this figure, the right line ag is a diameter, bf a chord, de 
a chord, ch a radius, and cm a radius, while the circular line de is an arc, 
bf an arc, and ag a semicircle. The portion of the surface included between 
an arc and its chord, as de, is a segment, that between two chords, as bdef 
a 2;o?ie, that between two radii and their arc, as chm, a sector, while a sector 
of 90°, as acm is a quadrant. 

Fig. 19. J.?i Equilateral Triangle 2s a plain figure with three equal sides. 

On a horizontal line lay off 1| in. for the base of the triangle. Since 
the sides are equal, the angles must be equal. The sum of the angles in a 
triangle is always equal to two right angles or 180°, one-third of 
which is 60°. Hence, Hues drawn from the extremities of the base with the 
60° Triangle will intersect to produce an equilateral triangle. 



16 JUNIOR COUESE. 



f 



Fig. 20. An Isosceles Triangle has two of its sides equal. 

Draw a horizontal base Ij in., from each extremity of which, as a 
centre, describe an arc of 2J in. radius. The intersection of these arcs 
will be the apex of the Isosceles Triangle. 

Fig. 21. A Scalene Triangle has all of its sides unequal. } 

Draw a horizontal base IJ in., from one extremity of which describe j, 

an arc Ij in. radius, and from the other, an arc 2 J in. radius. Lines J 

connecting these extremities with the intersection of the arcs will complete the ' 

Scalene Triangle. 

Fig. 22. A Eight-Angled Triangle has one right angle (90°). 
Draw a horizontal base If in., at one extremity of which erect a 
perpendicular, 2 in. The line completing the triangle is the hypothenuse. 

Fig. 23. A Square is a plane figure with four equal sides and four equal 
angles, each 90°. 

Draw a horizontal base 1| in., and a perpendicular side 1} in. 
Draw the other sides parallel, respectively, with this base and side to com- 
plete the square. 

Fig. 24. A Rhombus has four equal sides, but unequal angles. 
Opposite angles are equal and opposite sides parallel. 

Let the length of the sides be IJ in. and the inclination be J in. 
Draw a horizontal base li in. At a distance of A an in. from one 



1 



MECHANICAL DRAWING. 17 

end of the base, erect a perpendicular, and from this end of the base, with a 
radius of IJ in., describe an arc cutting the perpendicular. Through this 
intersection draw a horizontal line for the side opposite the base. Draw an 
inclined line from the intersection to one end of the base and from the other 
end draw a parallel line cutting the side opposite the base. This will com- 
plete the E-hombus. 

Fig. 25. A Rectangle has four equal angles (90°). 
Draw a rectangle IJ in. wide and 2 in. high. 

Fig. 26. A Rhomboid has its opposite sides equal, but its angles not right 
angles. 

Let the sides be 2 and IJ in., and the inclination of the short side 
be i in. 

Draw a horizontal line and erect a 2-in. perpendicular to it. From 
the foot of the perpendicular describe an arc IJ in. radius. Draw a second 
horizontal line J in. above the first. Draw a line from the intersection of 
the arc with the second horizontal line to the foot of the perpendicular, 
and a parallel line from the top of the perpendicular. A second perpendicular, 
drawn from the same intersection, will complete the Rhombus. 

Fig. 27. Given a circle, to inscribe or circumscribe a Pentagon or plane' 
figure with five equal sides. Draw a 2J-in. circle, and with the spacing 
dividers, set by guess to a distance approximately one and one-sixth times the 
2 



18 JUNIOR COUESE. 

radius, step around the circle, beiug careful to keep exactly on the 
circumference and to not prick the paper. Judge by the result whether 
to increase or diminish this distance. Continue the trial until the fifth step 
coincides with the starting-point. Then step around finally, making prick 
marks which can be seen. Lines connecting these prick marks will form an 
inscribed Pentagon and tangents to the circle at these points (as in Fig. 11) 
will form a circumscribed Pentagon. 

Fig. 28. To inscribe and circumscribe a Hexagon or plane figure with 
six equal sides. Draw three diameters at angles of 60° with each other. 
Lines connecting the extremities of these diameters will form the inscribed 
Hexagon. Draw six tangents to the circle with the T-square and 30° 
Triangle to form the circumscribed Hexagon. 

The diameter of the circle is commonly called the diameter across corners 
or long diameter of the inscribed hexagon and the diameter across flats or 
short diameter of the circumscribed hexagon. 

Fig. 29. To inscribe or circumscribe a Heptagon or plane figure with 
seven equal sides. 

Divide the circle, as in Fig. 27, by stepping seven sides. 

Fig. 30. To inscribe or circumscribe an Octagon or plane figure with eight 
equal sides. 

Proceed as in Fig. 28, but using the 45° Triangle. 



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MECHANICAL DRAWING. 19 

PLATE 3. 

Fig. 31. Given the length ab, or major axis, and width de, or minor 
axis, to construct an Ellipse. 

An Ellipse is a curve generated by a point moving in a plane so that 
the sum of the distances of this moving point from two fixed points shall be 
constantly equal. 

To understand this, drive two tacks in a board, to each of which fasten 
one end of a string of any length greater than the distance between the tacks, 
and press a piece of chalk or pencil against the string to keep it taut. The 
pencil is the generating point, and the length of the string is the sum of its 
distances from the two fixed points [the tacks] called the foci. Travel the pencil 
around on the board, guided by the string, and it will describe an Ellipse. 
A moment's thought will show that the length of the string is equal to the 
length of the Ellipse, or the major axis, therefore if, when the major and 
minor axes are given, we describe arcs from the extremities of the minor axis 
of radius equal to half of the major axis, the intersections of these arcs with 
the major axis will give us the foci. If, now, we describe arcs from each focus 
with any portion of the major axis as radius and intersect these by arcs from 
each opposite focus with the remaining portion of the major axis as radius, 
the intersection of these arcs will be points of the Ellipse. 

On this principle, describe an Ellipse 4 in. long and 2J in. wide. 



20 



JUNIOR COUKSE. 



Intersect the major axis by arcs of 2 in. radius, struck from d and e, 
to obtain the foci ff. With any portion of ah as radius, as ag, describe arcs 
from / and / and, with the remaining portion, hg, as radius, intersect these 
arcs. Proceed with other portions of ab as radius until enough intersections 
are obtained. Set the Irregular Curve to coincide with as many of these as 
possible, and draw a curved line through them. As the Ellipse is symmetrical 
about the two axes, make slight pencil marks on the Irregular Curve to include 
the portion of its edge just used and transfer these marks to opposite sides 
in order to identify the portion required for making three other similar parts 
of the Ellipse. Then shift the Irregular Curve to correspond with other 
intersections, and so on. A small portion of the ends of the Ellipse can be 
drawn with the compasses to insure symmetry. 

To facilitate the use of the Irregular Curve it is a good plan to make 
marks about J in. apart along the edges on both sides, exactly opposite 
each other, and number each fourth mark consecutively, but the same on both 
sides. This enables the same portion of the curve to be readily used in 
different places, or to be reversed for symmetrical work. 

Fig. 32. To describe an Oval or approximate Ellipse, by means of circular 
arcs J when the length and width are given. 

Let the length ab, be 4 in., and the width de, be 2 J in. 

Lay off half the width,=cc?, from o;ie end of the length, =6/. The 
-remaining portion of the length will be the radius, gh, for the top and bottom 



MECHANICAL DRAWING. 21 

of the oval. From / toward h lay off one-half of fc=^Jk, Then hb will be 
the radius of the ends of the oval. 

The majority of the Ellipses, which occur in draughting, can be approxi- 
mated by this method with sufficient accuracy for practical purposes. 

Fig. 33. A Parabola is a cmwe generated by a j)oint moving in a plane 
so that its distance from a given point shall be constantly equal to its distance 
from a given line. 

The given point, /, is the focus, the given line, cc?, the directrix, the line 
through the focus perpendicular to the directrix, vf is the axis, and the inter- 
section of the curve with the axis, v, is the apex. 

To draw a Parabola with the focus, /, at a distance of 1 in. from the 
directrix, cd. Locate the vertex, v, at one- half the distance between the focus 
and directrix. Draw a series of lines parallel to the directrix. With the 
focus as centre, cut each of these lines with an arc of radius equal to the 
distance of the line from the directrix and the intersections will be points of 
the curve. Draw the curve as in Fig. 31. 

Fig. 34. To construct a Parabola when the height, or abscissa, vb, and the 
width, or twice the ordinate aby are given. 

Draw a rectangle of the given height, 2J in., and width, 3J in. 
Divide the base into any number of equal parts and erect perpendiculars at 
these points. Divide the side of the rectangle into the same number of equal 



22 



JUNIOR COURSE. 



parts and draw lines from these points to the vertex. Where each of these 
lines intersects the corresponding perpendicular will be a point of the curve. 

Fig. 35. There are certain Rolled Curves, which are useful in mechanics, 
and an understanding of which is important. 

A Cycloid is a curve generated by a point on the circumference of a circle 
rolling upon a straight line without slipping. 

Draw a horizontal base line and above it a circle 2 in. diameter and 
below it one 4 in. diameter, the centres being on the same perpendicular. 
Let the upper circle be rolled to the left and the lower to the right. Draw 
a horizontal line through the centre of each circle to indicate the paths of 
these centres. With the spacing dividers, step equal divisions, say -f§ in., 
along the base line, starting at the vertical centre line. At each of these divis- 
ions draw perpendiculars to intersect the horizontal centre lines. From each 
of these intersections describe portions of the original circles, which will rep- 
resent their positions after having rolled to each successive division. With 
the spacing dividers unaltered and starting from the divisions on the base lines, 
step back on each corresponding circle the number of divisions it has moved 
from the original position, or, in other words, measure on the circumference of 
the rolling circle in its successive positions the length of the portion of the base 
line on which it has rolled. By using small divisions the error arising from the 
difference between arc and chord will be inappreciable. 



MECHANICAL DRAWING. 23 

Fig. 36. An Epicycloid is a curve generated by a point on the circum- 
fei^ence of a circle rolling on the outside of another circle without slipping y and 
a Hypocycloid is that generated by rolling on the inside. 

On a vertical centre-line draw a portion of a base-circle 8 in. diameter, 
and, tangent to it on the same centre-line, draw an outside and an inside rolling 
or generating-circle 2 in. diameter. From the centre of the base-circle describe 
arcs passing through the centres of the rolling-circles to show the paths of 
these centres. From the point of tangency step equal divisions, say ^ in., 
on the base-circle and from the centre of the latter draw radial lines through 
these divisions intersecting the paths of centres. From each of these intersec- 
tions describe portions of the rolling-circle and on them step off the original 
divisions in the same manner as in Fig. 35. 

Fig. 37. This is the same as Fig. 36, excepting that the base-circle is 
4 in. diameter, the rolling-circle remaining 2 in. It illustrates a curious 
and important fact, namely, that when the diameter of the Rolling- Circle is one-half 
that of the Base-Circle^ the Hypocycloid becomes a straight line, a radius. 

Fig. 38. An Involute is a curve generated by a point on a straight 
line rolling on a circle without slipping, or, more clearly, it is the cuy^ve described 
by the end of a taut string as it is unwound from a circle. 

Draw a circle 2 in. diameter and, starting from any desired point, 
step off equal divisions, -f^ in., on the circumference. Draw tangents at 



24 



JUNIOE COUESE. 



each of these divisions and from the point of tangency step oif on each tangent 
the original division as many times as the point is distant from the starting-point 
or, in other words, measure on each tangent the length of the arc of which it is 
the development. 

Construct also the Involute of a circle 4 in. diameter. 



PLATE 4. 

Heretofore we have been treating entirely of points, lines and surfaces. We 
have seen that points have no dimensions, neither length, breadth nor thickness, 
that lines have only length and that surfaces have only length and breadth. We 
have seen that the points, lines, and surfaces, which we have considered, lie in 
one plane and can be fully determined and expressed by one view on the paper. 

It now becomes necessary for us to express Solids in such manner that their 
form and dimensions shall be so fully determined that the solid object itself can 
be constructed without any further description or explanation than that given in 
the draAving. 

This is the sole purpose of Mechanical Drawing, to give such an illus- 
tration of the required object as to enable it to be accurately and definitely built 
from the drawing and the drawing alone, and not to make a picture or repre- 
sentation- of the object as it would appear in nature. 

As all solid objects have length, breadth, and thickness, and as only two of 



MECHANICAL DRAWING. 25 

these dimensions can be represented in exact form and size on one plane or view, 
it is necessary to use more than one view for the complete determination of all solids. 
As the paper, on which the drawing is made, lies in one plane only, it is nec- 
essary to devise a clear, simple, logical, and uniform method of representing 
the other planes or of drawing the various views, so that the different parts 
of the object will be shown in form and size and in their proper relation to 
each other, in order that the workman can accurately construct it without 
any verbal directions. 

Fig. 39. Let it he required to make a Mechanical Drawing of a Rectan- 
gular Prism 2^ in. high, 2 in. wide, and 1 in. thick. 

We know that this has six sides, the opposite ones being equal rect- 
angles ; hence, to show the shape of the different surfaces of the prism it 
is necessary to draw three rectangles. How shall we arrange three rectangles 
to make clear what they represent and to prevent any misunderstanding of 
the meaning of the drawing? The best method for this and the one used by 
the most progressive American and English draughtsmen is based upon the 
following theory. 

The object is supposed to be surrounded by planes at a short distance 
from it, the planes being perpendicular to each other. From each point of 
the object, perpendicular to each of these planes, lines are supposed to be 
projected, and the points of intersection of these perpendicular lines with these 
planes form the Projections of the object. One of these planes is supposed 



28 JUNIOR COURSE. 

to be the plane of the paper and all the other planes to be revolved about 
their intersections with this plane until thej coincide with it, thus bringing 
them all into the plane of the paper. The lines of intersection of the planes 
with each other are called the Axes of Fi^ojection. They are the axes or 
hinges, as it were, about which the planes are supposed to be swung. 

To understand this clearly draw upon a square piece of card- board a 
horizontal line, MN^ and a vertical line, vth'\ to represent these axes of pro- 
jection. Cut out the corner, NtN'y and fold the card on the lines hi and vt 
into positions at right angles with the original. By standing this on its 
lower edges upon a table it will be seen that vth and vth' are vertical planes 
perpendicular to each other, and that hth" is a horizontal plane perpendicular 
to both the others, that the axis vt is vertical, the axis ht horizontal, and 
that Nt and h"t coincide and are horizontal. The object to be drawn, a 
rectangular prism in this instance, is supposed to be upon the table surrounded 
by these planes. It will, therefore, be behind vth, to the left of vtN, and 
below MN'y and the projections drawn on vth will represent a front view, 
called the Front Elevation, those on vth' a right-hand side view, called 
the Side Elevation, and those on hth" a top view, called the Plan. If 
the card is then flattened out and laid upon the drawing-board it will 
be seen how two of the planes are supposed to be revolved upon the axes 
to bring them into the one plane of the drawing-paper. It will also be 
noted that the top view comes above the front view, the right-hand side view 



MECHANICAL DRAWING. 27 

to the right, and the left-hand side view (if one were necessary) would come 
to the left, thus making the clearest, most logical, and most convenient relative 
arrangement. 

To proceed with the drawing of the rectangular prism, draw a rectangle 
2J in. high and 2 in. wide. This will be the Front Elevation. 
At a short distance above this, say f in., draw the horizontal line ht, which 
will represent the edge of a horizontal Plane of Projection or more properly 
its intersection with the vertical plane of projection, called its trace on that 
plane, and will be the horizontal axis of projection. The drawing thus far 
represents the projection of the prism on a vertical plane in front of it, and 
shows a horizontal plane above it. Now draw lines, with long-and-short dot, 
perpendicular to ht to represent the vertical Projecting Lines from the prism 
which intersect the horizontal plane of projection. Fix upon the distance 
which the prism is supposed to be behind the vertical plane, say f in., 
imagine the horizontal plane revolved up into the plane of the paper (or in 
line with the vertical plane) and continue the projecting lines indefinitely. 
The projection of the prism on this horizontal plane will be a rectangle, the 
ends of which will be part of the projecting lines already drawn, the front 
of which will be a line parallel to ht and f in. behind it, and the back 
of which will be another horizontal parallel line at a distance of 1 in., 
the thickness of the prism, from the first. This will complete the Plan. 

These two views are all that are really necessary for a simple object like 



28 JUNIOR COURSE. 

this prism, as the length, breadth, and thickness are fully determined by them, 
but it is best to make three views in all our studies of principles, because 
in actual practice very few drawings of structures having any complication 
can be made to give all necessary information by means of two views only. 

To draw the third view or Side Elevation imagine a vertical plane } 
in. to the right of the prism, draw its trace vt, and imagine it revolved 
into the plane of the paper. Then vt will be the vertical and th' the hori- 
zontal axis of projection, and th' will be the same as tN'. To prove that ih' 
and th'' are different views of the same line, cut the card-board model of 
the planes of the projection on the line vi and hinge th' to th", when the 
planes w^ill fold into the original box-shape, and will unfold to bring the 
plane vth' to the right of hth" instead of to the right of vth as before; 
or the Side Elevation will come to the right of the Plan instead of to the 
right of the Front Elevation, and will be at right angles to its former 
position, an arrangement of the views which often proves convenient. 

Having now determined the side vertical plane vth', draw horizontal pro- 
jecting lines from the Front Elevation indefinitely to the right and from the 
Plan to the axis th". With iJ as a centre, describe arcs indicating the revolu- 
tion of the side vertical plane, and drop vertical projecting lines from these 
arcs to intersect the horizontal ones already drawn. The lines included between 
these intersections will be the projection of the Prism upon a vertical plane 
perpendicular to the former one and will complete the Side Elevation. 



MECHANICAL DRAWING. 29 

A thorough understanding of these p7Hnciples, ivhich are the vei^y foundation 
of correct mechanical drawing, is very important, and this Figure should not 
be passed until it is obtained. 

The next step is to ink the drawing, and no mechanical drawing is 
complete, or anything more than a sketch, until inked. 

We have already seen that in drawing a simple prism lines have been 
used which do not represent any part of the prism, but merely indicate im- 
aginary planes and lines of projection, and it is evident that if all these 
lines were inked in the same manner, the result would be confusing and the 
effect inartistic. The first object being to bring the shape and size of the 
prism out prominently and clearly, its projections should be inked black. 
Hence the rule: — All lines representing parts of the Object should be black, 
those representing visible parts being full lines and those representing hidden parts 
being dotted lines. 

The clearness, beauty, and realism of the drawing are increased by mak- 
ing certain lines heavy after a conventional mode, which, although not scien- 
tific, has proven to be the most efficient and to require the least mental effort 
in selection. These heavy lines are called Shade Lines. The scientific 
method, which is universally taught and never practiced after the first blush 
of apprenticeship, indicates by shade lines the shadows which would be pro- 
duced by rays of light falling upon the actual object at angles of 45° 
with both the horizontal and vertical planes, the direction being from the 



left front. This distributes the shade lines in the several views in the fol- 
lowing manner : — In the Front Elevation the right-hand and lower edges ; in 
the Plan, the right-hand and upper edges ; in the right Side Elevation the 
right-hand and lower edges, and in the left Side Elevation, the left-hand 
and lower edges. 

In the conventional method, the right-hand and lower edges are made 
shade lines in all the views j thus enabling the process to soon become a mere 
matter of habit, requiring but little thought, while the scientific method re- 
quires constant care to avoid placing them incorrectly, because of their not 
being in the same relative position in all the views. As all the drawings 
in the accompanying Plates are shade-lined by the conventional method, at- 
tention to them will probably be of more service than any further descrip- 
tion. One rule, however, should always be borne in mind, — Never make the 
line of intersection of two planes, of which both can be seen, a shade line. 

On the subject of inking, the following rules should be observed : 

Always ink the black lines first. 

Always ink all the circles and curves before the straight lines, because they 
are generally tangent to straight lines, and it is easier to draw a straight 
line correctly tangent to a curve than the reverse. 

The lower right-hand quadrant of exterior, and the upper left-hand quad- 
rant of interior curved edges should be shaded with the heavy part gradually 
blended into the fine part. 



MECHANICAL DRAWING. 31 

Always ink the fine straight lines next after the curves by setting the pen 
to the desired degree of fineness and making all the fine lines of the draw- 
ing without altering the pen, in order to produce the effect of uniformity. 

Last of all, open the pen and make all the shade lines of one thickness. 

All dotted lines are black and fine. 

The black lines being all completed, ink next the axes of projection, centre 
lines, and any important bases or lines of reference which do not represent 
any part of the object but are desired as explanatory. These lines should be 
Blue, but can be red. 

Next ink the dimension lines, projecting lines, and any construction lines 
used in obtaining the lines of the object, the preservation of which is desirable. 
These lines should be Red. 

The Arrow-heads or Index Points at the extremities of the dimension-lines 
should be black, made with the ivriting pen, and should always be in actual 
contact with the line to which they "point, because they indicate definite size 
and not merely direction. 

Always write the dimensions in line with and never inclined or perpendicular 
to the dimension-line. 

Never make an inclined line between the numerator and denominator of a 
fraction, as the practice is apt to cause mistakes in reading the dimension. 

For all lettering on a draioing, adopt som.e style of printing or conven- 
tional script, and never use your ordinary handwriting. 



32 JUNIOR COUESE, 

All Blue and all Red lines should he finer than the fine Mack lines, in 
order to avoid giving them prominence. 

The lines which have been directed to be blue may be red, but in a 
drawing of much detail it adds to the clearness to make them blue. 

Fig. 40. This is the same as Fig. 39, except that the prism has an 
opening through its thickness If in. high and IJ in. wide, leaving 
walls f in. thick. As this opening is invisible in the Plan and Side 
Elevation, its projections in these views are shown by black dotted lines, 
which, as already explained, are used for indicating invisible lines of the 
object. 

Fig. 41. Although a mechanical drawing of any required object is 
always made with the planes of projection parallel to the main features of 
the object, yet, in designing large structures, it frequently occurs that one 
integral part is at an angle with another, and it thus becomes important 
to understand the theory and be proficient in the practice of making pro- 
jections on planes which are not parallel with these main features. 

In this figure it is required to draw the same object as in the last, but 
with its front making an angle of 30° with the front vertical plane of 
projection, as shown. Draw the Plan complete at the required angle. Lay off 
on the front vertical plane the outside and inside heights and project these 
across to the side vertical plane. Draw projecting lines from all the points in 



MECHANICAL DRAWING. 33 

the Plan to both the vertical planes to locate the position of these points on 
the indefinite horizontal projecting lines already drawn. In inking this fig- 
ure, be careful to dot all the lines which are hidden. 

Fig. 42. Draw the same object with its front inclined at an angle of 
45° with the vertical plane and in the opposite direction from that in the 
last figure. 

As this figure completes the first plate of projections, the student, before 
going to the next, should draw other rectangular prisms of different dimen- 
sions and twisted on the horizontal plane to different angles, in order to 
become familiar with the principles. 

PLATE 5. 

Fig. 43. To maJce a mechanical drawing of a Wedge 2 J in. high^ with 
a base 2 iii. long and 1 in. luide. 

Draw, in the Plan, the rectangle of the base and the line of intersection 
of the two inclined sides, which line will coincide, in this view, with the 
centre line of the base. Draw the projecting lines to the Front and Side Ele- 
vations. Lay off the perpendicular height of the wedge on the vertical pro- 
jecting line which passes through the centre of the base in the Side Elevation 
and complete this view by drawing a horizontal projecting line for the base 
and connecting its extremities with the apex by inclined lines. Project the 
apex and the base across to complete the Front Elevation. 
3 



34 JUNIOR COXJHSE. 

Fig. 44. Draw the same Wedge with its base remaining horizontal , hut 
at an angle of 45° with the vertical planes of projection. 

An evident fact to be noted and always remembered is that if any point, 
line or solid is moved or twisted on a horizontal plane, the height of its projec- 
tions on the vertical planes will not be altered thereby. 

Fig. 45. Draw a Rectangular Pyramid with a vertical height of 2 J 
in. and a base 2 in, long and 1 in. wide. 

Fig. 46. Draw a Triangular Prism with sides 2 J in. long and 2 in- 
wide, one side to be in a horizontal plane. 

On the completion of this plate, the student should make another with the 
Pyramid of Fig. 45 twisted on the horizontal plane to diiferent angles and 
with another Pyramid of different proportions. 

PLATE 6. 

In order to acquire familiarity with the theory and a complete understand- 
ing of the meaning and relation of the three views in a drawing, we have 
heretofore drawn the axes of projection and have used them as bases to work 
from. They have served the purpose of lines of demarcation between the 
planes of projection and of constant reminders that the different views are projec- 
tions of the same object on different planes perpendicular to each other. We 
will now discontinue the actual use of these lines, although they must not be 



MECHANICAL DRAWING. 35 

lost sight of in imagination, but the fact must always be borne in mind that 
when one view of an object is projected directly from another view, the two 
planes of projection are always at right angles with each other, and that there 
is an intersection of these two planes which would be the Axis. 

Besides omitting the axis, we shall now abbreviate the Projecting Lines, 
making them merely sufficient to indicate the direction, and shall eventually 
omit them altogether, because it is desirable to have a drawing free of all 
lines which do not facilitate a comprehension of it. 

Fig. 47. Let it be required to draw a hollow Triangular Prism open at the 
ends, each side being 2 J in. long, 2 in. wide, and f in. thick, one of the sides 
being horizontal and at an angle of 30° icith the front vertical plane. 

Locate in the Plan a point for the position of the centre of the horizontal 
side of the prism, tli rough which point draw the horizontal line ah. This wall 
be the trace on the horizontal plane of projection of a vertical plane parallel 
to the front vertical plane of projection. [By the trace of a plane is meant 
its line of intersection with another plane.) At an angle of 30° with ab 
and passing through the central point already located, draw the centre line 
cd, which will represent a central vertical plane parallel with the edges of the 
prism. On this centre line complete the plan of the horizontal side of the 
prism. This will, of course, be its projection on a horizontal plane. We now 
want a vertical plane on which to draw the triangular end of the prism. The 
axis of such a plane must be parallel to the short side of the rectangle in the Plan 



36 JUNIOR COUESE. 

and the projection of this side on the plane must be a line also parallel. 
Hence, project this line, and on it construct the End Elevation of the 
prism. This End Elevation is the projection of the prism upon a vertical 
plane parallel with its end, called a vertical Auxiliary Plane, which is 
perpendicular to the horizontal plane, but at an angle of 60° to the front 
vertical plane of projection. 

From the End Elevation complete the Plan and project all the points 
down to the Front Elevation, obtaining their heights from the End Elevation. 

At any convenient distance to the right, draw a vertical line a'h' for the 
trace of the central vertical plane ah on the side vertical plane. Draw pro- 
jecting lines from the Front Elevation across a'b' and on them lay off from 
a'h' the distances of the points from ah in the Plan. Connect the proper points 
by lines to complete the Side Elevation. 

Close attention should be paid to this figure, as it is the first introduction 
to the method employed in practice. Note that the line ah is the Plan of a 
vertical plane and that a'h' is the Side Elevation of the same vertical plane, 
and that if any point is at a certain distance from ah in the Plan it will be 
at the same distance from a'h' in the Side Elevation. Note that ef is the 
trace of a vertical plane at right angles with ah, and that cd is the trace of 
still another vertical plane at an angle of 30° with ah. 

Fig. 48. To draio a Hexagonal Prism 2 J m. high and 2 in. across the 
flats. 



MECHANICAL DRAWING. 37 

In the Plan draw the traces of two vertical planes at right angles and 
about their intersection construct the Hexagon. Lay off the heiglit upon the 
Front Elevation, to which project the points from the Plan. In the Side 
Elevation draw the trace of a vertical plane, from which lay off the width of 
the Hexagon taken from the Plan. 

Fig. 49. Draio the same Hexagonal Prism, leaning 30° to the right, with 
one side parallel to the Front Elevation. 

As the prism is parallel to the Front Elevation it will appear there in its 
true size, hence we must commence with this view. Draw the centre line of 
the prism at the required angle, 30°, with a perpendicular passing through 
the centre of the base. On this centre line construct the Hexagon for the End 
Elevation, being careful to make one of its sides parallel with the front 
vertical plane according to the conditions given. Project the points from this 
to the Front Elevation, and from the latter complete the Plan and Side 
Elevation. 

Fig. 50. Draw a Pentagonal Prism 2J in. long, the inscribed circle of 
the Pentagon being 2 in. diameter, the Prism leaning 45° to the left and having 
one side parallel to the Front Elevation. 

Draw the centre line of the prism in the Front Elevation at the given 
angle, 45°, on which line construct the Pentagon, and complete the projections 
as before. 



38 JUNIOR COURSE. 

As an exercise, draw a plate containing four different prisms leaning in 
different directions at different angles. 

PLATE 7. 

Fig. 51. Draw a 2-in. Hexagonal Pyramid 2 J in. high, and find the 
exact shape and size of its inclined sides. 

Draw the plan and elevations as before. 

To find the shape and size of the inclined sides we can select either of 
them, because they are alike. We cannot project any of them from the Plan 
nor from the Front Elevation because they are inclined to both these planes 
of projection. In the Side Elevation, however, two of them are perpendicular 
to the plane of projection and we can select the side ab. We know that the 
point a is the projection of the apex, and the point b that of the base of this 
side. We also know that if we project the side ab upon a plane parallel to 
it and then revolve this plane into the plane of the paper we will obtain the 
real shape and size of this side. 

To do this, draw a line cd, parallel to ab, and at any convenient distance 
from it. Draw projecting lines, perpendicular to a6, from the apex a to cd 
aud from the base b across cd. On the latter projecting line lay off on each 
side of the centre line, cd, half the length of the base, obtained from the Plan. 
Connect these points with the projected apex to obtain the true size and shape 
of the side. 



MECHANICAL DBA WING. 39 

Fig. 52. Draw the same Pyramid truncated, or cut off, at a 'point on its 
axis IJ in, above the base and find the true shape and size of the surface 
left by the cut. 

Draw the plan and elevations of the entire pyramid as before and on the 
Front Elevation lay off a point on the axis IJ in. above the base, and 
through this point draw a line at 45° with the axis. Project the points 
where this line cuts the lines of intersection of the sides of the pyramid to 
the corresponding lines in the Plan and Side Elevation and complete these views 
of the truncated pyramid. 

To obtain the true shape and size of the cut surface, project it upon an 
auxiliary plane parallel to it, as in Fig. 51. This would fulfill the con- 
ditions given, but it is generally desirable to project the entire object on this 
auxiliary plane and not merely the surface whose true size is required. This 
makes a complete drawing of the truncated pyramid, the auxiliary view showing 
the cut surface in its true size and also in its relation to the rest of the pyramid. 

No difficulty need be experienced in drawing these projections as long as 
the fact is remembered that the centre line in the Plan, in the Side Elevation 
and in the Auxiliary View are all traces of one and the same vertical plane 
upon the planes of projections of these views and that any point in either of 
these views is at the same vertical distance from the centre line as in the 
others. 

Fig. 53. Draw the same Pyramid with two side.'i of its base perpendicular 



40 JUNIOR COURSE. 

to the Front Elevation and truncate it by a plane at an angle of 60° with 
the base and intersecting the axis IJ in. above the base and draw an 
auxiliary view parallel with the cut surface. 

Fig. 54. Draw a Pyramid the same as in Fig. 51, and truncate it by a 
plane parallel to and at a distance of 1 in. from the base. 

PLATE 8. 

Fig. 55. Draw a Pentagonal Pyramid 2J in. high, the inscribed circle 
of the base being 2 in. diameter and one side of the base being parallel 
with the Front Elevation. 

Fig. 56. To find the length of a line which is inclined to all the ^planes 
of projection. 

Take for example the line formed by the intersection of two of the in- 
clined sides of the Pentagonal Prism in Fig. 55, the one of which the Plan 
is ab, the Front Elevation a'6', and the Side Elevation a"b''. Copy these 
projections exactly. It is now necessary to project ab upon an auxiliary 
plane parallel to it. Draw through b' and b'' a horizontal line for the 
trace of a horizontal plane containing the point b, and draw parallel with 
ab a line for the trace of this same horizontal plane upon the auxiliary 
plane of projection, from which line lay off the perpendicular height of 
the point, af', above the horizontal plane, obtained from either the Front or 



MECHANICAL DRAWING. 41 

Side Elevation. Project h to b"\ and a"'h"' will be the line in its true 
length. 

Fig. 57. Draw a TriQ.ngular Pyramid 2 J in, high with each side of 
its base 2 J in. long, and one side making an angle of 15° with the front 
vertical plane, and draw one of the inclined sides in its true size and shape. 

After drawing the projections of the Pyramid project from the Plan the 
side abc upon a plane perpendicular to this side. The projection will be the 
line a"'b^''y the dotted line passing through b'" being the trace of the hori- 
zontal plane of the base on the vertical auxiliary plane of projection, and the 
point a'" being at the same vertical distance from this line as the vertical 
height of the Pyramid. Parallel with a'"b"' draw a centre line and project 
a'" upon it, as a"" , and from b''' project a line across it upon which lay 
off b""c"^' equal to 6c, then a^'"b""c"" will be the true shape and size of 
the side abc. 

Fig. 58. Copy the three views of the front side of the Pyramid in Fig. 
57, and from them find the true size and shape of this side. 



The Intermediate Course, which follows this, will treat of solids with 
curved surfaces, the intersections of solids and the development of their 
surfaces. 



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